Download Sequences and Modular Arithmetic and more Exams Discrete Mathematics in PDF only on Docsity!
Discrete Mathematics Final Exam Review
Questions and Answers | 100% Pass
Discrete Math for Computer Science Guaranteed | Graded A+ |
DISCRETE MATH Exam
Course Title and Number: DISCRETE MATH Exam Title: DISCRETE MATH Midterm and Final Exam Date: Midterm and Final Exam 2024- 2025 Instructor: [Insert Instructor’s Name] Student Name: [Insert Student’s Name] Student ID: [Insert Student ID]
Examination
180 minutes
Instructions:
**1. Read each question carefully.
- Answer all questions.
- Use the provided answer sheet to mark your** **responses.
- Ensure all answers are final before submitting the** **exam.
- Please answer each question below and click Submit** **when you have completed the Exam.
- This test has a time limit, The test will save and** **submit automatically when the time expires
- This is Exam which will assess your knowledge on** the course Learning Resources.
Good Luck!
Read All Instructions Carefully and Answer All the Questions Correctly Good Luck: - For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list.Assuming that your formula or rule is correct, determine the next three terms of the sequence. a) 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1,... b) 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8,... c) 1, 0, 2, 0, 4, 0, 8, 0, 16, 0,... d) 3, 6, 12, 24, 48, 96, 192,... e) 15, 8, 1, −6, −13, −20, −27,... f) 3, 5, 8, 12, 17, 23, 30, 38, 47,... g) 2, 16, 54, 128, 250, 432, 686,... h) 2, 3, 7, 25, 121, 721, 5041, 40321,... - Answer>> NA Look at Unit 2.6 - Answer>> NA
Suppose that a and b are integers, a ≡ 4 (mod 13), and b ≡ 9 (mod 13). Find the integer c with 0 ≤ c ≤ 12 such that a) c ≡ 9a (mod 13) b) c ≡ 11b (mod 13) c) c ≡ a + b (mod 13) d) c ≡ 2a + 3b (mod 13) e) c ≡ a2 + b2 (mod 13) f) c ≡ a3 − b3 (mod 13). - Answer>> NA
Evaluate these quantities. a) 13 mod 3 b) −97 mod 11 c) 155 mod 19 d) −221 mod 23 - Answer>> NA
Convert the octal expansion of each of these integers to a binary expansion. a) (572) sub 8 b) (1604)sub 8 c) (423)sub 8 d) (2417)sub 8 - Answer>> NA
Convert the hexadecimal expansion of each of these integers to a binary expansion. a) (80E)sub 16 b) (135AB)sub 16 c) (ABBA)sub 16 d) (DEFACED)sub 16 - Answer>> NA
Determine whether each of these integers is prime. a) 21 b) 29 c) 71 d) 97 e) 111 f) 143 - Answer>> NA
Show that log sub 2 3 is an irrational number. Recall that an irrational number is a real number x that cannot be written as the ratio of two integers. - Answer>> NA
What are the greatest common divisors of these pairs of integers? a) 3^7 · 5^3 · 7^3, 2^11 · 3^5 · 5^ b) 11 · 13 · 17, 2^9 · 3^7 · 5^5 · 7^ c) 23^31, 23^ d) 41 · 43 · 53, 41 · 43 · 53 e) 3^13 · 5^17, 2^12 · 7^
f) 1111, 0 - Answer>> NA
What is the least common multiple of each pair? a) 3^7 · 5^3 · 7^3, 2^11 · 3^5 · 5^ b) 11 · 13 · 17, 2^9 · 3^7 · 5^5 · 7^ c) 23^31, 23^ d) 41 · 43 · 53, 41 · 43 · 53 e) 3^13 · 5^17, 2^12 · 7^ f) 1111, 0 - Answer>> NA
Use the Euclidean algorithm to find a) gcd(12, 18). b) gcd(111, 201). c) gcd(1001, 1331). d) gcd(12345, 54321). e) gcd(1000, 5040). f) gcd(9888, 6060). - Answer>> NA
Prove that 1^2 + 3^2 + 5^2 +···+ (2n + 1)^2 = (n +
- (2n + 1)(2n + 3)/3 whenever n is a nonnegative integer. - Answer>> NA
a) Find a formula for the sum of the first n even positive integers. b) Prove the formula that you conjectured in part (a) - Answer>> NA
Prove that for every positive integer n, 1 · 2 + 2 · 3 +···+ n(n + 1) = n(n + 1)(n + 2)/3. - Answer>> NA
How many different three-letter initials can people have? - Answer>> NA
How many different three-letter initials are there that begin with an A? - Answer>> NA
A committee is formed consisting of one representative from each of the 50 states in the United States, where the representative from a state is either the governor or one of the two senators from that state. How many ways are there to form this committee? - Answer>> NA
How many license plates can be made using either two uppercase English letters followed by four digits or two digits followed by four uppercase English letters? - Answer>> NA
In how many ways can a photographer at a wedding arrange six people in a row, including the bride and groom, if a) the bride must be next to the groom? b) the bride is not next to the groom? c) the bride is positioned somewhere to the left of the groom? - Answer>> NA
List all the permutations of {a, b, c}. - Answer>> NA
How many permutations of {a, b, c, d, e, f, g} end with a? - Answer>> NA
Find the number of 5-permutations of a set with nine elements. - Answer>> NA
How many possibilities are there for the win, place, and show (first, second, and third) positions in a horse race with 12 horses if all orders of finish are possible? - Answer>> NA
A group contains n men and n women. How many ways are there to arrange these people in a row if the men and women alternate? - Answer>> NA
In how many ways can a set of five letters be selected from the English alphabet? - Answer>> NA
A coin is flipped 10 times where each flip comes up either heads or tails. How many possible outcomes a) are there in total? b) contain exactly two heads? c) contain at most three tails? d) contain the same number of heads and tails? - Answer>> NA Statistics It is the science of collecting, organizing, summarizing, and analyzing information to draw conclusion. Descriptive Statistics Collecting, summarizing, presenting and analyzing an entire set of collected data. Inferential Statistics
It is a set of rational numbers. Z It is a set of integers. W It is a set of whole numbers. N It is a set of natural numbers. R It is a set of real numbers. C It is a set of complex numbers. Mathematical Expressions It is the name given to a mathematical object of interest, which may be a quantity, number, and combinations of these using different operations. Mathematical Sentences It expresses complete mathematical thought about relation of mathematical object to another mathematical object. Precise, Concise, & Powerful What are the characteristics of Mathematical Language? Statement/Proposition It is a declarative sentence which can be regarded as true or false. Simple Statement It contains only one idea. Compound Statement It contains of two or more statement with connective. Quantified Statements It involve terms such as all, each, every, no, none, some, here exists, and at least one. Universal Quantifiers It is either include or exclude every element of the universal set. These includes all, each, every, no, and none. Existential Quantifiers
It claims the existence of something but don't include the entire universal set. These are some, there exists, at least one. Negation It is a corresponding statement with the opposite truth value. Truth Tables It displays the relationships between the truth values of statements or propositions. Conjuction The propositions of p and q is the compound statement "p and q" denoted as p ↑ q which is true only when both p and q are true, otherwise, is false. Disjunction The propositions of p and q is the compount statement "p and q" denoted as p ↓ q which is false only when both p and q are false, otherwise, is true. Conditional The propositions p and q is the compound statement "if p, then q." denotef by p → q which is false only when p is true and q is false. Biconditional The propositions p and q is the statement "p if and only if q." denoted as p ←→ q which is true only when both p and q have the same truth value. Negation The statement p is denoted by ~p (not p) where ~ is the symbol for "not." The truth vale of it is always the opposite of the truth value of the original statement. p, q, & r What lower case letters we use to represent the statement? Relation Symbols These are used for comparison and act as verbs in the mathematical language. Grouping Symbols
d) The summer in Maine is hot and sunny. - Answer>> a) Mei does not have an MP3 player b) There is pollution in New Jersey. c) 2 + 1 != 3 d) The summer in Maine is not hot and sunny.
Which of these sentences are propositions? What are the truth values of those that are propositions? a) Boston is the capital of Massachusetts. b) Miami is the capital of Florida. c) 2 + 3 = 5. d) 5 + 7 = 10. e) x + 2 = 11. f ) Answer this question. - Answer>> a) Yes, True b) Yes, False c) Yes, True d) Yes, False e) No f) No
Let p and q be the propositions "Swimming at the New Jersey shore is allowed" and "Sharks have been spotted near the shore," respectively. Express each of these compound propositions as an English sentence. a) ¬q b) p ∧ q c) ¬p ∨ q d) p → ¬q e) ¬q → p f) ¬p → ¬q g) p ↔ ¬q h) ¬p ∧ (p ∨ ¬q) - Answer>> NA
Let p and q be the propositions
p : It is below freezing. q : It is snowing. Write these propositions using p and q and logical connectives (including negations). a) It is below freezing and snowing. b) It is below freezing but not snowing. c) It is not below freezing and it is not snowing. d) It is either snowing or below freezing (or both). e) If it is below freezing, it is also snowing. f ) Either it is below freezing or it is snowing, but it is not snowing if it is below freezing. g) That it is below freezing. - Answer>> NA
Construct a truth table for each of these compound propositions. a) p → ¬q b) ¬p ↔ q c) (p → q) ∨ (¬p → q) d) (p → q) ∧ (¬p → q) e) (p ↔ q) ∨ (¬p ↔ q) f ) (¬p ↔ ¬q) ↔ (p ↔ q) - Answer>> NA
Translate the given statement into propositional logic using the propositions provided. You cannot edit a protected Wikipedia entry unless you are an administrator. Express your answer in terms of e: "You can edit a protected Wikipedia entry" and a: "You are an administrator." - Answer>> NA
Translate the given statement into propositional logic using the propositions provided.
Exercise relates to inhabitants of the island of knights and knaves created by Smullyan, where knights always tell the truth and knaves always lie. You encounter two people, A and B. Determine, if possible, what A and B are if they address you in the ways described. If you cannot determine what these two people are, can you draw any conclusions? A says "I am a knave orB is a knight" andB says nothing
- Answer>> NA
Exercise relates to inhabitants of the island of knights and knaves created by Smullyan, where knights always tell the truth and knaves always lie. You encounter two people, A and B. Determine, if possible, what A and B are if they address you in the ways described. If you cannot determine what these two people are, can you draw any conclusions? A says "We are both knaves" and B says nothing - Answer>> NA
Use a truth table to verify the distributive law p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r). - Answer>> NA
Show that each of these conditional statements is a tautology by using truth tables. a) (p ∧ q) → p b) p → (p ∨ q) c) ¬p → (p → q) d) (p ∧ q) → (p → q) e) ¬(p → q) → p f) ¬(p → q) → ¬q - Answer>> NA
Show that each conditional statement in Exercise 9 is a tautology without using truth tables. - Answer>> NA
Determine whether (¬q ∧ (p → q)) → ¬p is a tautology.
- Answer>> NA
Show that (p → q) ∧ (q → r) → (p → r) is a tautology. - Answer>> NA
Let P (x) be the statement "x spends more than five hours every weekday in class," where the domain for x consists of all students. Express each of these quantifications in English. a) ∃xP(x) b) ∀xP (x) c) ∃x ¬P (x) d) ∀x ¬P (x) - Answer>> NA
Translate these statements into English, where C(x) is "x is a comedian" and F (x) is "x is funny" and the domain consists of all people. a) ∀x(C(x) → F (x)) b) ∀x(C(x) ∧ F (x)) c) ∃x(C(x) → F (x)) d) ∃x(C(x) ∧ F (x)) - Answer>> NA
Determine the truth value of each of these statements if the domain consists of all integers. a) ∀n(n + 1 > n) b) ∃n(2n = 3n)
Express each of these statements by a simple English sentence. a) W(Sarah Smith, www.att.com) b) ∃xW(x, www.imdb.org) c) ∃yW(José Orez, y) d) ∃y(W(Ashok Puri, y) ∧ W(Cindy Yoon, y)) e) ∃y∀z(y = (David Belcher) ∧ (W(David Belcher, z) → W (y,z))) f) ∃x∃y∀z((x = y) ∧ (W (x, z) ↔ W (y, z))) - Answer>> NA
Let T (x, y) mean that student x likes cuisine y, where the domain for x consists of all students at your school and the domain for y consists of all cuisines. Express each of these statements by a simple English sentence. a) ¬T (Abdallah Hussein, Japanese) b) ∃xT (x, Korean) ∧ ∀xT (x, Mexican) c) ∃y(T (Monique Arsenault, y) ∨ T (Jay Johnson, y)) d) ∀x∀z∃y((x = z) → ¬(T (x, y) ∧ T (z, y))) e) ∃x∃z∀y(T (x, y) ↔ T (z, y)) f) ∀x∀z∃y(T (x, y) ↔ T (z, y)) - Answer>> NA
Let S(x) be the predicate "x is a student," F (x) the predicate "x is a faculty member," and A(x, y) the predicate "x has asked y a question," where the domain consists of all people associated with your school. Use quantifiers to express each of these statements. a) Lois has asked Professor Michaels a question. b) Every student has asked Professor Gross a question. c) Every faculty member has either asked Professor Miller a question or been asked a question by Professor Miller. d) Some student has not asked any faculty member a question.
e) There is a faculty member who has never been asked a question by a student. f ) Some student has asked every faculty member a question. g) There is a faculty member who has asked every other faculty member a question. h) Some student has never been asked a question by a faculty member. - Answer>> NA
Use quantifiers and predicates with more than one variable to express these statements. a) Every computer science student needs a course in discrete mathematics. b) There is a student in this class who owns a personal computer. c) Every student in this class has taken at least one computer science course. d) There is a student in this class who has taken at least one course in computer science. e) Every student in this class has been in every building on campus. f) There is a student in this class who has been in every room of at least one building on campus. g) Every student in this class has been in at least one room of every building on campus - Answer>> NA
Find the argument form for the following argument and determine whether it is valid. Can we conclude that the conclusion is true if the premises are true? If Socrates is human, then Socrates is mortal. Socrates is human. ∴ Socrates is mortal. - Answer>> NA
